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Chapter 2: Problem 2

The number \(J\), in thousands, of cans of frozen orange juice sold weekly is a function of the price \(P\), in dollars, of a can. In a certain grocery store, the formula is $$ J=11-2.5 P $$ a. Express using functional notation the number of cans sold weekly if the price of a can is \(\$ 1.40\), and then calculate that value. b. At what price will there be \(7.25\) thousand cans sold weekly? c. Solve for \(P\) in the formula above to obtain a formula expressing \(P\) as a function of \(J\). d. At what price will there be \(7.75\) thousand cans sold weekly?

Short Answer

Expert verified
a) 7.5 thousand cans. b) \$1.50. c) \(P = \frac{11 - J}{2.5}\). d) \$1.30.

Step by step solution

01

Using Functional Notation

Express the number of cans sold weekly as a function: \(J(P) = 11 - 2.5P\). To find the number of cans sold at a price of \(\$1.40\), substitute \(P = 1.40\) into the function: \(J(1.40) = 11 - 2.5 \times 1.40\).
02

Calculate the Value for Part (a)

Calculate the result: \(J(1.40) = 11 - 3.5 = 7.5\). Thus, when the price is \(\$1.40\), 7.5 thousand cans are sold weekly.
03

Set Up Equation for Part (b)

To find the price when 7.25 thousand cans are sold, set \(J = 7.25\) and solve: \(7.25 = 11 - 2.5P\).
04

Solve for P When J = 7.25

Rearrange the equation: \(2.5P = 11 - 7.25\). This simplifies to \(2.5P = 3.75\). Divide both sides by 2.5: \(P = 1.5\). So, \(\$1.50\) is the price for 7.25 thousand cans sold weekly.
05

Solve for P in terms of J for Part (c)

Start with \(J = 11 - 2.5P\). Rearrange to isolate \(P\): \(2.5P = 11 - J\). Therefore, \(P = \frac{11 - J}{2.5}\).
06

Find P When J = 7.75 for Part (d)

Given \(J = 7.75\), substitute into the function \(P = \frac{11 - 7.75}{2.5}\). Calculate \(P = \frac{3.25}{2.5} = 1.3\). Thus, the price is \(\$1.30\) for 7.75 thousand cans sold weekly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Functional Notation
Functional notation is a way to describe functions in mathematics. Simply put, a function is a relation where every input has a unique output. For example, in the relation given by the formula \( J = 11 - 2.5P \), \( J \) is expressed as a function of \( P \). This can be written using functional notation as \( J(P) = 11 - 2.5P \).
This notation helps us easily see how the quantity \( J \) changes with different values of \( P \).
  • It specifies the dependent variable \( J \) depends on the independent variable \( P \).
  • By substituting values into \( P \), we can quickly find the value of \( J \).
For instance, if we want to find how many cans are sold when the price \( P \) is \$1.40, we substitute \( P = 1.40 \) into the functional notation: \( J(1.40) = 11 - 2.5 \times 1.40 \).
This simplifies the process and highlights the dynamic relationship described by the function.
Solving Equations
Solving equations is about finding the value that makes the equation true. In our context, solving might entail determining either the number of cans sold (\( J \)) or the price (\( P \)) at which a certain number of cans are sold.
Let's consider a specific example with our function \( J(P) = 11 - 2.5P \). If you know \( J \) and want to find out \( P \), you rearrange the equation:
  • Set \( J = 11 - 2.5P \) when \( J \) is known.
  • Rearrange to isolate \( P \): \( 2.5P = 11 - J \).
  • Solve for \( P \) using division: \( P = \frac{11 - J}{2.5} \).
This strategy allows you to switch between knowing the price (and guessing the sales) and knowing the sales (and guessing the price).
Understanding this concept turns equations into powerful tools for interpreting and predicting relationships in data.
Price-Demand Relationship
The price-demand relationship represents how the price of an item affects the amount sold. In simple terms, it often shows that higher prices lead to lower demand, while lower prices might attract more buyers.
Our specific function \( J = 11 - 2.5P \) captures this notion perfectly:
  • "11" represents the theoretical maximum sales if the price is \\(0.
  • "-2.5" indicates the rate at which demand decreases as price increases. For every increase of \\)1, 2.5 thousand fewer cans are sold.
This relationship helps businesses strategize about pricing to optimize sales volume. Understanding how to express these situations as functions, and solve for unknowns, supports better decision-making in economics and marketing.
Observing how changes in price impact demand is crucial for setting competitive prices that maximize profit while maintaining customer interest.

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Most popular questions from this chapter

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